On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization
In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/727 |
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| author | A. A. Elsadany A. Aldurayhim H. N. Agiza Amr Elsonbaty |
| author_facet | A. A. Elsadany A. Aldurayhim H. N. Agiza Amr Elsonbaty |
| author_sort | A. A. Elsadany |
| collection | DOAJ |
| description | In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work. |
| first_indexed | 2024-03-11T09:34:31Z |
| format | Article |
| id | doaj.art-86c667273a4f4e3aa17824a9b4d35ff2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:34:31Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-86c667273a4f4e3aa17824a9b4d35ff22023-11-16T17:23:45ZengMDPI AGMathematics2227-73902023-02-0111372710.3390/math11030727On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and SynchronizationA. A. Elsadany0A. Aldurayhim1H. N. Agiza2Amr Elsonbaty3Department of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaDepartment of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, EgyptDepartment of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaIn this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work.https://www.mdpi.com/2227-7390/11/3/727complex cosine mapdiscrete fractionalfractal setsJulia set controlJulia sets synchronization |
| spellingShingle | A. A. Elsadany A. Aldurayhim H. N. Agiza Amr Elsonbaty On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization Mathematics complex cosine map discrete fractional fractal sets Julia set control Julia sets synchronization |
| title | On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization |
| title_full | On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization |
| title_fullStr | On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization |
| title_full_unstemmed | On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization |
| title_short | On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization |
| title_sort | on the fractional order complex cosine map fractal analysis julia set control and synchronization |
| topic | complex cosine map discrete fractional fractal sets Julia set control Julia sets synchronization |
| url | https://www.mdpi.com/2227-7390/11/3/727 |
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